ww Ud 
pee NT NX. NA UNT € mot 
MM WE NS "me SS 
— 
where, Q9 is the correlation matrix of class wi) 
which is defined as 
Q? 2 ExxIxe a?) (6) 
By combining (5) with the normal condition of 
bases luj9 ,..., Upi)9 | , 
Wud =1 | k=1,.p" (7) 
Using the Lagrange multiplier method, 
minimization of (2) is transformed to the 
minimization of next term (8) 
(D 
() K 
Su S00 -00wd- Sarin 
k=1 
k=1 J#i 
jl 
Taking the derivative of this term with respect to 
the base vectors uy (kK-—L..,p? ), we obtain 
necessary condition for minimizing solution. 
K 
oo Ou s Qum pi (9) 
ji 
fl 
From equation (9), it is know that the solution 
base vectors ua (k—...,p? ) of L9 is the eigen 
vectors of the next matrix. 
K 
Q- $9? - go (10) 
j*i 
pe 
In addition, setting the 1th eigen value of Qas 4 
X9, (8) becomes 
(0 (©) 
(i) 
Suro - pue ug BY (11) 
k=l k=1 k=1 
So, in order to minimize (8), we can select the 
eigen vectors which correspond to the minimum 
p? eigen values as the ortho-normal base of L(? ; 
Here, the dimension p® of subspace is the 
parameter to adjust the mean projection on the 
classes. 
Because the subspace L9 is uniquely 
determined from the base vectors wz? &=1,...,.p% ), 
the above procedure determines the subspaces to 
minimize the enhanced CLAFIC criterion (2). 
2.4 Unmixing by subspace method 
Once the class base vectors ug&9 (k-1,...,p? ) are 
determined as the eigen vectors corresponding to 
the eigen values of correlation matrix, projection 
matrix P? is calculated from equation (3). The 
length of the projection of the observed mixel 
spectral vector x on the class subspace .L& is 
calculated as, 
po 
x'poxz Yay (12) 
k=1 
783 
  
RE EEE A EE aq 
This projection length expresses how much of 
the mixel vector belongs to the class w@). By a 
natural extension of the membership values, we 
interpret this projection as a measure of the class 
component contained in the mixel vector and have 
defined the unmixing in each class as the 
projection on the class subspace calculated by (12). 
3. UNMIXING EXPERIMENT USING CASI 
IMAGE 
In order to check whether the unmixing by 
subspace method works effectively for hyper 
spectral images, we have conducted an unmixing 
experiment using a 288 channel CASI (Compact 
Airborne Spectral Imager) and compared the 
result with conventional statistical unmixing 
methods. 
3.1 Study site 
The spectral image used for our analysis is a 
CASI image acquired over the Kushiro wetland 
located in the north east Hokkaido Island, Japan 
(Figure. 1). The CASI spectral sensor can measure 
a spectrum from 470 to 920nm with a 1.8nm band 
width. The specification and the data acquisition 
conditions for the CASI sensor are shown in 
Table.1. The image was acquired at an altitude of 
3,000m by  Cesna404 aircraft. The ground 
resolution is longer (12.6m) in the aircraft flight. 
Each pixel in the image contains the mean 
spectral radiance ofthe ground target.. 
A selection of 7 bands from the original CASI 
image (spaced every 40 channels) is shown in the 
Figure.2. The first 4 channels are in visible 
spectrum and the others are in | near infra red. In 
the center of Figure 2 is Lake Akanuma and the 
artificial dike across the area is clearly visible. 
There are various wetland vegetation in this study 
area, especially reed, sedge and sedum is 
overlapping and continuously distributed over the 
sphagnum moss. 
Before the analysis, the CASI image was 
corrected for the geometric distortion caused by 
the rolling of the airplane and the digital numbers 
were converted to radiance values (Babey and 
Soffer, 1993). 
3.2 Unmixing 
The spectral characteristics of 7 land cover 
classes used for unmixing is shown in Figure 3. 
All the classes are wetland vegetation 
communities except for the road, and water classes. 
The spectral difference between these vegetation 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996